Theorem r0cld 23717

Description: The analogue of the T₁ axiom (singletons are closed) for an R₀ space. In an R₀ space the set of all points topologically indistinguishable from 𝐴 is closed. (Contributed by Mario Carneiro, 25-Aug-2015.)

Hypothesis

Ref	Expression
kqval.2	⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦})

Assertion

Ref	Expression
r0cld	⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴 ∈ 𝑋) → {𝑧 ∈ 𝑋 ∣ ∀𝑜 ∈ 𝐽 (𝑧 ∈ 𝑜 ↔ 𝐴 ∈ 𝑜)} ∈ (Clsd‘𝐽))

Distinct variable groups:   𝑥,𝑜,𝑦,𝑧,𝐴   𝑜,𝐽,𝑥,𝑦,𝑧   𝑜,𝐹,𝑧   𝑜,𝑋,𝑥,𝑦,𝑧

Allowed substitution hints:   𝐹(𝑥,𝑦)

Proof of Theorem r0cld

Step	Hyp	Ref	Expression
1		kqval.2	. . . . . 6 ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦})
2	1	kqffn 23704	. . . . 5 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐹 Fn 𝑋)
3	2	3ad2ant1 1134	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴 ∈ 𝑋) → 𝐹 Fn 𝑋)
4		fncnvima2 7009	. . . 4 ⊢ (𝐹 Fn 𝑋 → (◡𝐹 “ {(𝐹‘𝐴)}) = {𝑧 ∈ 𝑋 ∣ (𝐹‘𝑧) ∈ {(𝐹‘𝐴)}})
5	3, 4	syl 17	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴 ∈ 𝑋) → (◡𝐹 “ {(𝐹‘𝐴)}) = {𝑧 ∈ 𝑋 ∣ (𝐹‘𝑧) ∈ {(𝐹‘𝐴)}})
6		fvex 6849	. . . . . 6 ⊢ (𝐹‘𝑧) ∈ V
7	6	elsn 4583	. . . . 5 ⊢ ((𝐹‘𝑧) ∈ {(𝐹‘𝐴)} ↔ (𝐹‘𝑧) = (𝐹‘𝐴))
8		simpl1 1193	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴 ∈ 𝑋) ∧ 𝑧 ∈ 𝑋) → 𝐽 ∈ (TopOn‘𝑋))
9		simpr 484	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴 ∈ 𝑋) ∧ 𝑧 ∈ 𝑋) → 𝑧 ∈ 𝑋)
10		simpl3 1195	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴 ∈ 𝑋) ∧ 𝑧 ∈ 𝑋) → 𝐴 ∈ 𝑋)
11	1	kqfeq 23703	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → ((𝐹‘𝑧) = (𝐹‘𝐴) ↔ ∀𝑦 ∈ 𝐽 (𝑧 ∈ 𝑦 ↔ 𝐴 ∈ 𝑦)))
12		eleq2w 2821	. . . . . . . . 9 ⊢ (𝑦 = 𝑜 → (𝑧 ∈ 𝑦 ↔ 𝑧 ∈ 𝑜))
13		eleq2w 2821	. . . . . . . . 9 ⊢ (𝑦 = 𝑜 → (𝐴 ∈ 𝑦 ↔ 𝐴 ∈ 𝑜))
14	12, 13	bibi12d 345	. . . . . . . 8 ⊢ (𝑦 = 𝑜 → ((𝑧 ∈ 𝑦 ↔ 𝐴 ∈ 𝑦) ↔ (𝑧 ∈ 𝑜 ↔ 𝐴 ∈ 𝑜)))
15	14	cbvralvw 3216	. . . . . . 7 ⊢ (∀𝑦 ∈ 𝐽 (𝑧 ∈ 𝑦 ↔ 𝐴 ∈ 𝑦) ↔ ∀𝑜 ∈ 𝐽 (𝑧 ∈ 𝑜 ↔ 𝐴 ∈ 𝑜))
16	11, 15	bitrdi 287	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → ((𝐹‘𝑧) = (𝐹‘𝐴) ↔ ∀𝑜 ∈ 𝐽 (𝑧 ∈ 𝑜 ↔ 𝐴 ∈ 𝑜)))
17	8, 9, 10, 16	syl3anc 1374	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴 ∈ 𝑋) ∧ 𝑧 ∈ 𝑋) → ((𝐹‘𝑧) = (𝐹‘𝐴) ↔ ∀𝑜 ∈ 𝐽 (𝑧 ∈ 𝑜 ↔ 𝐴 ∈ 𝑜)))
18	7, 17	bitrid 283	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴 ∈ 𝑋) ∧ 𝑧 ∈ 𝑋) → ((𝐹‘𝑧) ∈ {(𝐹‘𝐴)} ↔ ∀𝑜 ∈ 𝐽 (𝑧 ∈ 𝑜 ↔ 𝐴 ∈ 𝑜)))
19	18	rabbidva 3396	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴 ∈ 𝑋) → {𝑧 ∈ 𝑋 ∣ (𝐹‘𝑧) ∈ {(𝐹‘𝐴)}} = {𝑧 ∈ 𝑋 ∣ ∀𝑜 ∈ 𝐽 (𝑧 ∈ 𝑜 ↔ 𝐴 ∈ 𝑜)})
20	5, 19	eqtrd 2772	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴 ∈ 𝑋) → (◡𝐹 “ {(𝐹‘𝐴)}) = {𝑧 ∈ 𝑋 ∣ ∀𝑜 ∈ 𝐽 (𝑧 ∈ 𝑜 ↔ 𝐴 ∈ 𝑜)})
21	1	kqid 23707	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐹 ∈ (𝐽 Cn (KQ‘𝐽)))
22	21	3ad2ant1 1134	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴 ∈ 𝑋) → 𝐹 ∈ (𝐽 Cn (KQ‘𝐽)))
23		simp2 1138	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴 ∈ 𝑋) → (KQ‘𝐽) ∈ Fre)
24		simp3 1139	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ 𝑋)
25		fnfvelrn 7028	. . . . . 6 ⊢ ((𝐹 Fn 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝐹‘𝐴) ∈ ran 𝐹)
26	3, 24, 25	syl2anc 585	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴 ∈ 𝑋) → (𝐹‘𝐴) ∈ ran 𝐹)
27	1	kqtopon 23706	. . . . . . 7 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) ∈ (TopOn‘ran 𝐹))
28	27	3ad2ant1 1134	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴 ∈ 𝑋) → (KQ‘𝐽) ∈ (TopOn‘ran 𝐹))
29		toponuni 22893	. . . . . 6 ⊢ ((KQ‘𝐽) ∈ (TopOn‘ran 𝐹) → ran 𝐹 = ∪ (KQ‘𝐽))
30	28, 29	syl 17	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴 ∈ 𝑋) → ran 𝐹 = ∪ (KQ‘𝐽))
31	26, 30	eleqtrd 2839	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴 ∈ 𝑋) → (𝐹‘𝐴) ∈ ∪ (KQ‘𝐽))
32		eqid 2737	. . . . 5 ⊢ ∪ (KQ‘𝐽) = ∪ (KQ‘𝐽)
33	32	t1sncld 23305	. . . 4 ⊢ (((KQ‘𝐽) ∈ Fre ∧ (𝐹‘𝐴) ∈ ∪ (KQ‘𝐽)) → {(𝐹‘𝐴)} ∈ (Clsd‘(KQ‘𝐽)))
34	23, 31, 33	syl2anc 585	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴 ∈ 𝑋) → {(𝐹‘𝐴)} ∈ (Clsd‘(KQ‘𝐽)))
35		cnclima 23247	. . 3 ⊢ ((𝐹 ∈ (𝐽 Cn (KQ‘𝐽)) ∧ {(𝐹‘𝐴)} ∈ (Clsd‘(KQ‘𝐽))) → (◡𝐹 “ {(𝐹‘𝐴)}) ∈ (Clsd‘𝐽))
36	22, 34, 35	syl2anc 585	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴 ∈ 𝑋) → (◡𝐹 “ {(𝐹‘𝐴)}) ∈ (Clsd‘𝐽))
37	20, 36	eqeltrrd 2838	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴 ∈ 𝑋) → {𝑧 ∈ 𝑋 ∣ ∀𝑜 ∈ 𝐽 (𝑧 ∈ 𝑜 ↔ 𝐴 ∈ 𝑜)} ∈ (Clsd‘𝐽))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3052  {crab 3390  {csn 4568  ∪ cuni 4851   ↦ cmpt 5167  ^◡ccnv 5625  ran crn 5627   “ cima 5629   Fn wfn 6489  ‘cfv 6494  (class class class)co 7362  TopOnctopon 22889  Clsdccld 22995   Cn ccn 23203  Frect1 23286  KQckq 23672

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5304  ax-pr 5372  ax-un 7684

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5521  df-xp 5632  df-rel 5633  df-cnv 5634  df-co 5635  df-dm 5636  df-rn 5637  df-res 5638  df-ima 5639  df-iota 6450  df-fun 6496  df-fn 6497  df-f 6498  df-f1 6499  df-fo 6500  df-f1o 6501  df-fv 6502  df-ov 7365  df-oprab 7366  df-mpo 7367  df-map 8770  df-qtop 17466  df-top 22873  df-topon 22890  df-cld 22998  df-cn 23206  df-t1 23293  df-kq 23673

This theorem is referenced by:  nrmr0reg  23728